{
  "cells": [
    {
      "cell_type": "markdown",
      "id": "f89242ce",
      "metadata": {},
      "source": [
        "# Gal3 IMU EKF Tutorial\n",
        "\n",
        "This tutorial shows how to use GTSAM’s Gal3-specific EKF (Gal3ImuEKF) with synthetic IMU data and occasional position measurements. You’ll learn:\n",
        "- How the predict step integrates IMU into a Gal3 increment and propagates covariance.\n",
        "- How to write a position measurement in the EKF’s local coordinates (the correct H = [0_3x1, 0_3x1, R_3x1, 0]).\n",
        "- How to interpret the results with ±2σ uncertainty bands for yaw/pitch/roll, velocity, and position.\n",
        "\n",
        "We simulate a simple motion and run the EKF end-to-end, then visualize estimates vs. ground truth along with uncertainty."
      ]
    },
    {
      "cell_type": "markdown",
      "id": "license_cell",
      "metadata": {
        "tags": [
          "remove-cell"
        ]
      },
      "source": [
        "GTSAM Copyright 2010-2022, Georgia Tech Research Corporation,\nAtlanta, Georgia 30332-0415\nAll Rights Reserved\n\nAuthors: Frank Dellaert, et al. (see THANKS for the full author list)\n\nSee LICENSE for the license information"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "f4724d5a",
      "metadata": {},
      "source": [
        "<a href=\"https://colab.research.google.com/github/borglab/gtsam/blob/develop/python/gtsam/examples/Gal3ImuEKFExample.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "id": "e1b272ef",
      "metadata": {
        "tags": [
          "remove-cell"
        ]
      },
      "outputs": [],
      "source": [
        "# Install GTSAM from pip if running in Google Colab\n",
        "try:\n",
        "    import google.colab  # type: ignore\n",
        "    %pip install --quiet gtsam-develop\n",
        "except Exception:\n",
        "    pass  # Not in Colab"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "id": "fdd02b60",
      "metadata": {
        "tags": [
          "hide-input"
        ]
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import plotly.graph_objects as go\n",
        "import pandas as pd\n",
        "\n",
        "import gtsam\n",
        "from gtsam import Gal3, Rot3\n",
        "from gtsam import ConstantTwistScenario, ScenarioRunner\n",
        "\n",
        "# Helper to display yaw/pitch/roll in degrees\n",
        "def rot3_ypr_rad(R: Rot3):\n",
        "    \"\"\"Return yaw, pitch, roll (degrees) for a Rot3.\"\"\"\n",
        "    return np.degrees(R.ypr())"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "41612211",
      "metadata": {},
      "source": [
        "## Background and setup\n",
        "\n",
        "State: Gal3 X = (R, v, p, t) with rotation R ∈ SO(3), velocity v ∈ R^3, position p ∈ R^3, time t ∈ R.\n",
        "- EKF local coordinates are [δθ, δv_body, δp_body, δt], i.e., position/velocity increments are expressed in the body frame.\n",
        "- For a world-position measurement z ≈ p_world, the linearized measurement Jacobian is H = [0, 0, R, 0].\n",
        "\n",
        "We’ll now define a simple scenario that provides ground truth and noise-free IMU to drive the predict step."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "id": "a46d4dc2",
      "metadata": {},
      "outputs": [],
      "source": [
        "# --- Scenario: camera orbiting a fixed point ---\n",
        "radius = 30.0\n",
        "angular_velocity = np.pi  # rad/sec (half-turn per second)\n",
        "w_b = np.array([0, 0, angular_velocity])  # body yaw rate\n",
        "v_n = np.array([radius * angular_velocity, 0, 0])  # world-frame velocity\n",
        "\n",
        "scenario = ConstantTwistScenario(w_b, v_n)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "id": "9c6085bd",
      "metadata": {},
      "outputs": [],
      "source": [
        "# Simulation parameters\n",
        "dt = 1.0 / 180.0  # 1 degree per step\n",
        "T = 4.0           # total duration (s)\n",
        "N = int(T / dt)\n",
        "\n",
        "# Example of modeling an IMU using NED coordinates as the navigation frame \n",
        "# with sensor measurements in the body/sensor frame\n",
        "params = gtsam.PreintegrationParams.MakeSharedD(9.81)  # gravity in m/s^2\n",
        "\n",
        "# IMU covariances (tune as needed)\n",
        "params.setAccelerometerCovariance(np.diag([1e-3] * 3))\n",
        "params.setIntegrationCovariance(np.diag([1e-3] * 3))  # integration noise\n",
        "params.setGyroscopeCovariance(np.diag([1e-4] * 3))\n",
        "\n",
        "runner = ScenarioRunner(scenario, params, dt,\n",
        "                         gtsam.imuBias.ConstantBias(np.zeros(3), np.zeros(3)))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "id": "675a9ec7",
      "metadata": {
        "tags": [
          "hide-input"
        ]
      },
      "outputs": [
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                      "#FF97FF",
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                    ],
                    "font": {
                      "color": "#2a3f5f"
                    },
                    "geo": {
                      "bgcolor": "white",
                      "lakecolor": "white",
                      "landcolor": "#E5ECF6",
                      "showlakes": true,
                      "showland": true,
                      "subunitcolor": "white"
                    },
                    "hoverlabel": {
                      "align": "left"
                    },
                    "hovermode": "closest",
                    "mapbox": {
                      "style": "light"
                    },
                    "paper_bgcolor": "white",
                    "plot_bgcolor": "#E5ECF6",
                    "polar": {
                      "angularaxis": {
                        "gridcolor": "white",
                        "linecolor": "white",
                        "ticks": ""
                      },
                      "bgcolor": "#E5ECF6",
                      "radialaxis": {
                        "gridcolor": "white",
                        "linecolor": "white",
                        "ticks": ""
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                    },
                    "scene": {
                      "xaxis": {
                        "backgroundcolor": "#E5ECF6",
                        "gridcolor": "white",
                        "gridwidth": 2,
                        "linecolor": "white",
                        "showbackground": true,
                        "ticks": "",
                        "zerolinecolor": "white"
                      },
                      "yaxis": {
                        "backgroundcolor": "#E5ECF6",
                        "gridcolor": "white",
                        "gridwidth": 2,
                        "linecolor": "white",
                        "showbackground": true,
                        "ticks": "",
                        "zerolinecolor": "white"
                      },
                      "zaxis": {
                        "backgroundcolor": "#E5ECF6",
                        "gridcolor": "white",
                        "gridwidth": 2,
                        "linecolor": "white",
                        "showbackground": true,
                        "ticks": "",
                        "zerolinecolor": "white"
                      }
                    },
                    "shapedefaults": {
                      "line": {
                        "color": "#2a3f5f"
                      }
                    },
                    "ternary": {
                      "aaxis": {
                        "gridcolor": "white",
                        "linecolor": "white",
                        "ticks": ""
                      },
                      "baxis": {
                        "gridcolor": "white",
                        "linecolor": "white",
                        "ticks": ""
                      },
                      "bgcolor": "#E5ECF6",
                      "caxis": {
                        "gridcolor": "white",
                        "linecolor": "white",
                        "ticks": ""
                      }
                    },
                    "title": {
                      "x": 0.05
                    },
                    "xaxis": {
                      "automargin": true,
                      "gridcolor": "white",
                      "linecolor": "white",
                      "ticks": "",
                      "title": {
                        "standoff": 15
                      },
                      "zerolinecolor": "white",
                      "zerolinewidth": 2
                    },
                    "yaxis": {
                      "automargin": true,
                      "gridcolor": "white",
                      "linecolor": "white",
                      "ticks": "",
                      "title": {
                        "standoff": 15
                      },
                      "zerolinecolor": "white",
                      "zerolinewidth": 2
                    }
                  }
                },
                "title": {
                  "text": "Scenario Trajectory (Top View: X vs Y, colored by time)"
                },
                "xaxis": {
                  "title": {
                    "text": "X (m)"
                  }
                },
                "yaxis": {
                  "scaleanchor": "x",
                  "scaleratio": 1,
                  "title": {
                    "text": "Y (m)"
                  }
                }
              }
            }
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "times=np.linspace(0.0,T,N+1); states=[scenario.navState(t) for t in times]; positions=np.array([s.position() for s in states]); rotations=[s.attitude() for s in states]\n",
        "fig=go.Figure(); fig.add_trace(go.Scatter(x=positions[:,0],y=positions[:,1],mode='markers+lines',name='Scenario Trajectory',line=dict(width=2,color='black'),marker=dict(size=4,color=times[:positions.shape[0]],colorscale='Gray',colorbar=dict(title='Time (s)'))))\n",
        "scale=6.0; arrow_skip=45\n",
        "for i in range(0,len(positions),arrow_skip):\n",
        "    R=rotations[i].matrix(); pos=positions[i,:2]; f=R[:2,0]; r=R[:2,1]\n",
        "    fig.add_annotation(x=pos[0]+scale*f[0],y=pos[1]+scale*f[1],ax=pos[0],ay=pos[1],xref='x',yref='y',axref='x',ayref='y',showarrow=True,arrowhead=2,arrowsize=1,arrowwidth=1.5,arrowcolor='red',standoff=0,startstandoff=0)\n",
        "    fig.add_annotation(x=pos[0]+scale*r[0],y=pos[1]+scale*r[1],ax=pos[0],ay=pos[1],xref='x',yref='y',axref='x',ayref='y',showarrow=True,arrowhead=2,arrowsize=1,arrowwidth=1.5,arrowcolor='green',standoff=0,startstandoff=0)\n",
        "fig.update_layout(title='Scenario Trajectory (Top View: X vs Y, colored by time)',xaxis_title='X (m)',yaxis_title='Y (m)',margin=dict(l=0,r=0,b=0,t=40),yaxis=dict(scaleanchor='x',scaleratio=1)); fig.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "7a9fab50",
      "metadata": {},
      "source": [
        "## Simulate IMU and run the EKF\n",
        "\n",
        "We integrate IMU in the predict step and occasionally update with a position measurement. The position Jacobian must be H = [0, 0, R, 0] because EKF local coordinates use body-frame increments for p and v."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "id": "7698e840",
      "metadata": {},
      "outputs": [],
      "source": [
        "# Initialize EKF with ground-truth state and modest covariance\n",
        "X0: Gal3 = scenario.gal3(0.0)\n",
        "P0 = np.eye(10) * 0.1  # std ~ 0.316 on each component\n",
        "P0[9, 9] = 1e-6  #small variance for time \n",
        "ekf = gtsam.Gal3ImuEKF(X0, P0, params)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "id": "022045d8",
      "metadata": {},
      "outputs": [],
      "source": [
        "times = np.linspace(0.0, T, N + 1)\n",
        "\n",
        "# Storage for plotting\n",
        "ypr_true, ypr_est = [], []\n",
        "vel_true, vel_est = [], []\n",
        "pos_true, pos_est = [], []\n",
        "time_true, time_est = [], []\n",
        "\n",
        "# Std-dev storage from EKF covariance (for ±2σ bands)\n",
        "rot_std_deg_list = []  # yaw/pitch/roll std in degrees\n",
        "pos_std_list = []      # x/y/z std in meters\n",
        "vel_std_list = []      # vx/vy/vz std in m/s\n",
        "time_std_list = []\n",
        "\n",
        "# Simulate and filter\n",
        "Xk = X0\n",
        "for k, t in enumerate(times):\n",
        "    # Ground truth state from scenario\n",
        "    X_true: Gal3 = scenario.gal3(t)\n",
        "    ypr_true.append(rot3_ypr_rad(X_true.attitude()))\n",
        "    vel_true.append(np.asarray(X_true.velocity()))\n",
        "    pos_true.append(np.asarray(X_true.position()))\n",
        "    time_true.append(X_true.time())\n",
        "\n",
        "    if k == 0:\n",
        "        # Record initial estimate and covariance\n",
        "        X_est = ekf.state()\n",
        "        ypr_est.append(rot3_ypr_rad(X_est.attitude()))\n",
        "        vel_est.append(np.asarray(X_est.velocity()))\n",
        "        pos_est.append(np.asarray(X_est.position()))\n",
        "        time_est.append(X_est.time())\n",
        "\n",
        "        P = ekf.covariance()\n",
        "        std = np.sqrt(np.diag(P))\n",
        "        rot_std_deg_list.append(np.rad2deg(std[0:3]))\n",
        "        vel_std_list.append(std[3:6])\n",
        "        pos_std_list.append(std[6:9])\n",
        "        time_std_list.append(std[9])\n",
        "        continue\n",
        "\n",
        "    # Predict using noise-free IMU from scenario (at interval start)\n",
        "    omega_meas = runner.actualAngularVelocity(t - dt)\n",
        "    acc_meas = runner.actualSpecificForce(t - dt)\n",
        "    ekf.predict(omega_meas, acc_meas, dt)\n",
        "\n",
        "    # Log estimate after predict\n",
        "    X_est = ekf.state()\n",
        "    ypr_est.append(rot3_ypr_rad(X_est.attitude()))\n",
        "    vel_est.append(np.asarray(X_est.velocity()))\n",
        "    pos_est.append(np.asarray(X_est.position()))\n",
        "    time_est.append(X_est.time())\n",
        "\n",
        "    # Log current covariance std-devs\n",
        "    P = ekf.covariance()  # post-predict, pre-update\n",
        "    std = np.sqrt(np.diag(P))\n",
        "    rot_std_deg_list.append(np.rad2deg(std[0:3]))\n",
        "    vel_std_list.append(std[3:6])\n",
        "    pos_std_list.append(std[6:9])\n",
        "    time_std_list.append(std[9])\n",
        "\n",
        "    # Position measurement every M steps\n",
        "    M = 30  # measurement cadence\n",
        "    if k % M == 0:\n",
        "        predicted_position = X_est.position()\n",
        "        measured_position = X_true.position()\n",
        "        R = np.eye(3) * 1.0  # measurement covariance (tune as needed)\n",
        "\n",
        "        # H maps local error [δθ, δv_b, δp_b, δt] to position residual.\n",
        "        # Measurement Jacobian in EKF local coords: H = [0, 0, R, t]\n",
        "        H = np.zeros((3, 10))\n",
        "        H[:, 6:9] = X_est.attitude().matrix()\n",
        "        ekf.updateWithVector(predicted_position, H, measured_position, R)\n",
        "\n",
        "# Convert to arrays for plotting\n",
        "ypr_true = np.unwrap(np.vstack(ypr_true), axis=0)\n",
        "ypr_est = np.unwrap(np.vstack(ypr_est), axis=0)\n",
        "vel_true = np.vstack(vel_true)\n",
        "vel_est = np.vstack(vel_est)\n",
        "pos_true = np.vstack(pos_true)\n",
        "pos_est = np.vstack(pos_est)\n",
        "time_true = np.array(time_true)\n",
        "time_est = np.array(time_est)\n",
        "\n",
        "rot_std_deg = np.vstack(rot_std_deg_list)\n",
        "pos_std = np.vstack(pos_std_list)\n",
        "vel_std = np.vstack(vel_std_list)\n",
        "time_std = np.vstack(time_std_list)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "844223ec",
      "metadata": {},
      "source": [
        "## Plot Yaw/Pitch/Roll (degrees)\n",
        "\n",
        "We show estimated angles with ±2σ bands and dashed ground truth. Ideally, the truth lies mostly within the shaded bands if covariances are well tuned."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "id": "b16a8666",
      "metadata": {
        "tags": [
          "hide-input"
        ]
      },
      "outputs": [
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                    "dtype": "f8"
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                },
                {
                  "line": {
                    "color": "#ff0000",
                    "width": 2
                  },
                  "name": "Yaw est",
                  "type": "scatter",
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",
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                    "dash": "dash",
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                  "line": {
                    "color": "#0000ff",
                    "width": 2
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                    "dash": "dash",
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                      "x": 0.05
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                },
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                    "text": "Time (s)"
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                "yaxis": {
                  "title": {
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      ],
      "source": [
        "# YPR (deg) with ±2σ bands, RGB colors, dashed GT\n",
        "fig = go.Figure(); color_rgb = {0: ('rgba(255,0,0,0.15)', '#ff0000'), 1: ('rgba(0,128,0,0.15)', '#008000'), 2: ('rgba(0,0,255,0.15)', '#0000ff')}; names = {0: 'Yaw', 1: 'Pitch', 2: 'Roll'}\n",
        "for i in range(3):\n",
        "    fill_rgba, line_color = color_rgb[i]; mean = ypr_est[:, i]; std2p = 2.0 * rot_std_deg[:, i]; upper = mean + std2p; lower = mean - std2p\n",
        "    fig.add_scatter(x=times, y=upper, line=dict(color=line_color, width=0), showlegend=False)\n",
        "    fig.add_scatter(x=times, y=lower, fill='tonexty', fillcolor=fill_rgba, line=dict(color=line_color, width=0), showlegend=False)\n",
        "    fig.add_scatter(x=times, y=mean, name=f\"{names[i]} est\", line=dict(color=line_color, width=2))\n",
        "    fig.add_scatter(x=times, y=ypr_true[:, i], name=f\"{names[i]} true\", line=dict(color=line_color, dash='dash', width=2))\n",
        "fig.update_layout(title='YPR (deg) with ±2σ bands', xaxis_title='Time (s)', yaxis_title='Degrees'); fig.show()"
      ]
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    {
      "cell_type": "markdown",
      "id": "fca1d37b",
      "metadata": {},
      "source": [
        "## Plot Velocity (m/s)\n",
        "\n",
        "Velocity often benefits from frequent updates (e.g., GPS Doppler). Here we only have occasional position updates; expect velocity uncertainty to grow between updates and shrink slightly after them."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "id": "9d0e25cc",
      "metadata": {
        "tags": [
          "hide-input"
        ]
      },
      "outputs": [
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                    "dash": "dash",
                    "width": 2
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                  "name": "vx true",
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                },
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                    "text": "Time (s)"
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                "yaxis": {
                  "title": {
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      "source": [
        "# Velocity (m/s) with ±2σ bands\n",
        "fig = go.Figure(); color_rgb = {0: ('rgba(255,0,0,0.15)', '#ff0000'), 1: ('rgba(0,128,0,0.15)', '#008000'), 2: ('rgba(0,0,255,0.15)', '#0000ff')}\n",
        "for i, comp in enumerate(['x','y','z']):\n",
        "    fill_rgba, line_color = color_rgb[i]; mean = vel_est[:, i]; std2p = 2.0 * vel_std[:, i]; upper = mean + std2p; lower = mean - std2p\n",
        "    fig.add_scatter(x=times, y=upper, line=dict(color=line_color, width=0), showlegend=False)\n",
        "    fig.add_scatter(x=times, y=lower, fill='tonexty', fillcolor=fill_rgba, line=dict(color=line_color, width=0), showlegend=False)\n",
        "    fig.add_scatter(x=times, y=mean, name=f\"v{comp} est\", line=dict(color=line_color, width=2))\n",
        "    fig.add_scatter(x=times, y=vel_true[:, i], name=f\"v{comp} true\", line=dict(color=line_color, dash='dash', width=2))\n",
        "fig.update_layout(title='Velocity (m/s) with ±2σ bands', xaxis_title='Time (s)', yaxis_title='m/s'); fig.show()"
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      "source": [
        "## Plot Position (m)\n",
        "\n",
        "Position updates should pull the estimate toward truth and reduce the ±2σ band. If divergence occurs, check:\n",
        "- H matches EKF local coords ([0, 0, R, 0] for world-position).\n",
        "- Measurement noise R reflects your sensor.\n",
        "- Process noise (IMU) is not overconfident."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "id": "2ea60e82",
      "metadata": {
        "tags": [
          "hide-input"
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      },
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                      "bgcolor": "#E5ECF6",
                      "caxis": {
                        "gridcolor": "white",
                        "linecolor": "white",
                        "ticks": ""
                      }
                    },
                    "title": {
                      "x": 0.05
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                    "xaxis": {
                      "automargin": true,
                      "gridcolor": "white",
                      "linecolor": "white",
                      "ticks": "",
                      "title": {
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                  "text": "Position (m) with ±2σ bands"
                },
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                  "title": {
                    "text": "Time (s)"
                  }
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                "yaxis": {
                  "title": {
                    "text": "m"
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          },
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      ],
      "source": [
        "# Position (m) with ±2σ bands\n",
        "fig = go.Figure(); color_rgb = {0: ('rgba(255,0,0,0.15)', '#ff0000'), 1: ('rgba(0,128,0,0.15)', '#008000'), 2: ('rgba(0,0,255,0.15)', '#0000ff')}\n",
        "for i, comp in enumerate(['x','y','z']):\n",
        "    fill_rgba, line_color = color_rgb[i]; mean = pos_est[:, i]; std2p = 2.0 * pos_std[:, i]; upper = mean + std2p; lower = mean - std2p\n",
        "    fig.add_scatter(x=times, y=upper, line=dict(color=line_color, width=0), showlegend=False)\n",
        "    fig.add_scatter(x=times, y=lower, fill='tonexty', fillcolor=fill_rgba, line=dict(color=line_color, width=0), showlegend=False)\n",
        "    fig.add_scatter(x=times, y=mean, name=f\"p{comp} est\", line=dict(color=line_color, width=2))\n",
        "    fig.add_scatter(x=times, y=pos_true[:, i], name=f\"p{comp} true\", line=dict(color=line_color, dash='dash', width=2))\n",
        "fig.update_layout(title='Position (m) with ±2σ bands', xaxis_title='Time (s)', yaxis_title='m'); fig.show()"
      ]
    }
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